1
What Are Parameters?
•
Consider some probability distributions:
Ber(p)
Poi(
l
)
Multinomial(p
1
, p
2
, .
.., p
m
)
Uni(a, b)
Normal(
m
,
2
)
Etc.
•
Call these “parametric models”
•
Given model, parameters yield actual distribution
Usually refer to parameters of distribution as
Note that
that can be a vector of parameters
= p
=
l
= (p
1
, p
2
, .
.., p
m
)
= (a, b)
= (
m
,
2
)
Why Do We Care?
•
In real world, don’t know “true” parameters
But, we do get to observe data
o
E.g., number of times coin comes up heads, lifetimes of disk
drives produced, number of visitors to web site per day, etc.
Need to estimate model parameters from data
“Estimator” is random variable estimating parameter
•
Want “point estimate” of parameter
Single value for parameter as opposed to distribution
•
Estimate of parameters allows:
Better understanding of process producing data
Future predictions based on model
Simulation of processes
Recall Sample Mean
•
Consider
n
I.I.D. random variables X
1
, X
2
, .
.. X
n
X
i
have distribution
F
with E[X
i
] =
m
and Var(X
i
) =
2
We call sequence of X
i
a
sample
from distribution
F
Recall sample mean:
where
Recall variance of sample mean:
Clearly, sample mean X is a random variable
n
i
i
n
X
X
1
]
[
X
E
n
X
2
)
(
Var
Sampling Distribution
•
Note that sample mean X is random variable
“Sampling distribution of mean” is the distribution of
the random variable X
Central Limit Theorem tells us sampling distribution of
X is approximately normal when sample size,
n
, is
large
o
Rule of thumb for “large”
n
:
n
> 30, but larger is better (> 100)